92 lines
2.3 KiB
Text
92 lines
2.3 KiB
Text
import logic
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inductive nat : Type :=
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zero : nat,
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succ : nat → nat
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open eq
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namespace nat
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definition add (x y : nat)
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:= nat.rec x (λ n r, succ r) y
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infixl `+`:65 := add
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theorem add_zero_left (x : nat) : x + zero = x
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:= refl _
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theorem add_succ_left (x y : nat) : x + (succ y) = succ (x + y)
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:= refl _
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definition is_zero (x : nat)
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:= nat.rec true (λ n r, false) x
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theorem is_zero_zero : is_zero zero
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:= eq_true_elim (refl _)
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theorem not_is_zero_succ (x : nat) : ¬ is_zero (succ x)
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:= eq_false_elim (refl _)
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theorem dichotomy (m : nat) : m = zero ∨ (∃ n, m = succ n)
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:= nat.rec
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(or.intro_left _ (refl zero))
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(λ m H, or.intro_right _ (exists_intro m (refl (succ m))))
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m
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theorem is_zero_to_eq (x : nat) (H : is_zero x) : x = zero
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:= or.elim (dichotomy x)
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(assume Hz : x = zero, Hz)
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(assume Hs : (∃ n, x = succ n),
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exists_elim Hs (λ (w : nat) (Hw : x = succ w),
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absurd H (eq.subst (symm Hw) (not_is_zero_succ w))))
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theorem is_not_zero_to_eq {x : nat} (H : ¬ is_zero x) : ∃ n, x = succ n
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:= or.elim (dichotomy x)
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(assume Hz : x = zero,
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absurd (eq.subst (symm Hz) is_zero_zero) H)
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(assume Hs, Hs)
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theorem not_zero_add (x y : nat) (H : ¬ is_zero y) : ¬ is_zero (x + y)
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:= exists_elim (is_not_zero_to_eq H)
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(λ (w : nat) (Hw : y = succ w),
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have H1 : x + y = succ (x + w), from
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calc x + y = x + succ w : {Hw}
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... = succ (x + w) : refl _,
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have H2 : ¬ is_zero (succ (x + w)), from
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not_is_zero_succ (x+w),
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subst (symm H1) H2)
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inductive not_zero (x : nat) : Prop :=
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intro : ¬ is_zero x → not_zero x
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theorem not_zero_not_is_zero {x : nat} (H : not_zero x) : ¬ is_zero x
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:= not_zero.rec (λ H1, H1) H
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theorem not_zero_add_right [instance] (x y : nat) (H : not_zero y) : not_zero (x + y)
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:= not_zero.intro (not_zero_add x y (not_zero_not_is_zero H))
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theorem not_zero_succ [instance] (x : nat) : not_zero (succ x)
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:= not_zero.intro (not_is_zero_succ x)
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variable dvd : Π (x y : nat) {H : not_zero y}, nat
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variables a b : nat
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set_option pp.implicit true
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opaque add
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check dvd a (succ b)
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check (λ H : not_zero b, dvd a b)
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check (succ zero)
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check a + (succ zero)
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check dvd a (a + (succ b))
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transparent add
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check dvd a (a + (succ b))
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opaque add
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check dvd a (a + (succ b))
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transparent add
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check dvd a (a + (succ b))
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end nat
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